---

Statement

Notation (p. 125): $f$ is a monic polynomial (1), $E(f)$ the set where $|f|<1$, $\bar E(f)$ its closure and $D$ the open unit disk.

Problem 9 (p. 142). "Let $N(f)$ denote the number of components of $\bar E(f)$ which have diameter greater than 1, and let $N_n$ be the greatest value which $N(f)$ can assume, for polynomials (1) of degree $n$ with all $z_\nu$ in $D$. Is the sequence $\{N_n\}$ bounded?"

Added in proof (p. 148). The paper answers the question: "The sequence $\{N_n\}$ in Problem 9 is not bounded." Its construction moves the zeros $e^{\pm i\pi/n}$ of $z^n+1$ a short distance $\delta$ along the unit circle toward $z=1$. The $[(n-1)/2]$ leaves of the rosette $\bar E$ in the left half-plane then separate, while the other leaves merge, and for small $\delta$ each resulting component of $\bar E$ has diameter greater than $2^{1/n}-\varepsilon$; so $N_n\ge n/2$.

Problem 9 prints $D$, the open disk, while the construction places every zero on the unit circle and the follow-up question speaks of "the restriction that $|z_\nu|\leq1$" (p. 148); the construction answers the question for zeros in the closed disk. That reading is this page's.

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; Problem 9 on p. 142, the note added in proof on p. 148. The copy read is identified on the source card.

Read depth. Claims checked: the problem and the note were read clause by clause on the page images of pp. 142 and 148 on 2026-10-08, the disk symbol in Problem 9 at high resolution. The construction was read, not checked. Nothing here is independently reviewed.

Dependencies

The perturbation of $z^n+1$ resembles that of Theorem 7, which moves the same two zeros all the way to $1$.

Bears on

• #511: the problem is the question that follows this note, the p. 148 question, which drops the restriction on the zeros and raises the diameter threshold above $1$; the note's construction concerns the threshold $1$ only.